Sunday, October 11, 2009

10th Assignment: Write a review about the Windowed Fourier Transform. What are the most common windows? How is the Brillouin plane of this transform?

On this assignment, I will write a little bit about the Windowed Fourier Transform. As powerful as the Fourier transform may be, it has a major issue. Because it is originally designed to quantify the energy of all frequencies on a signal, it always takes under consideration the entire signal. It means that the results do not allow us to take any conclusion on when the frequencies have reached the peak or any other time-related information.All one can find by using Fourier Transform to analyze a signal is what frequencies are present on the signal and how much of each frequency was found. Windowed Fourier Transform adds a new constraint to the Fourier Transform in order to try reaching a certain level of time-related information for each frequency.

The core idea of the Windowed Fourier Transform is the addition of a window function to the traditional Fourier Transform. This Time Window function will segment the signal and will turn into 0 (zero) all values that do not belong to the specified time-window. From the notation perspective, this is what one will have:

As T is the time windows that one wants to study:

+T/2 + ∞
X(w) = ∫ x(t)e-jwt dt = ∫ w(t) x(t) e-jwt dt
-T/2 - ∞

with

w(t) = 1 when |t|<=T/2
and
w(t) = 0 when |t| > T/2

There are ups and downs about this method. By using time windows instead of the entire signal, the local maximum not necessarily represents the signal maximum. However, assuming that for the specific study, knowing only the local frequencies would be enough, this could be an interesting and effective way to locate the studied frequencies inside a time period.

How you choose your time-window is also important. Different time-windows will have different impacts on your results. Many different window functions have been proposed over time, each with its own advantages and disadvantages relative to the others. Here is a list with the most common time window functions: